Two concentric circles have radii $15\text{cm}$ (outer) and $9\text{cm}$ (inner). Find: (a) the area of the ring formed between them; (b) the length of the arc of the outer circle subtended by a central angle of $45^{\circ }$; (c) the length of the chord of the inner circle that subtends a central angle of $30^{\circ }$. Give all answers rounded to two decimal places.

Step-by-step solution

1. Step 1: (a) Area of ring = area of outer circle – area of inner circle.
2. Step 2: Compute outer area: $A_{\text{outer}}=\pi r_{\text{outer}}^2=\pi \times 15^2=\pi \times 225$.
3. Step 3: Compute inner area: $A_{\text{inner}}=\pi r_{\text{inner}}^2=\pi \times 9^2=\pi \times 81$.
4. Step 4: Subtract: $A_{\text{ring}}=\pi (225-81)=\pi \times 144$.
5. Step 5: Use $\pi =\frac{22}{7}$: $A_{\text{ring}}=\frac{22}{7}\times 144=\frac{3168}{7}=452.5714286$. Step 6: Round to two decimals: $452.57{\text{cm}}^2$. Step 7: (b) Arc length formula: $L=\frac{\theta }{360^{\circ }}\times 2\pi r$ with $r=15$ cm and $\theta =45^{\circ }$. Step 8: Compute fraction $\frac{45}{360}=\frac{1}{8}$. Step 9: Compute $2\pi r=2\times \frac{22}{7}\times 15=\frac{44}{7}\times 15=\frac{660}{7}=94.2857143$. Step 10: Multiply: $L=\frac{1}{8}\times 94.2857143=11.7857143$. Step 11: Round to two decimals: $11.79\text{cm}$. Step 12: (c) Chord length formula: $c=2rsin(\frac{\theta }{2})$ with $r=9$ cm and $\theta =30^{\circ }$. Step 13: Compute $\frac{\theta }{2}=15^{\circ }$. Step 14: Evaluate $sin15^{\circ }\approx 0.258819045$. Step 15: Multiply: $c=2\times 9\times 0.258819045=18\times 0.258819045=4.65874281$. Step 16: Round to two decimals: $4.66\text{cm}$.

Concept tested: Area of annulus, arc length, chord length

Published 7 October 2026 · How this solution was prepared and checked